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Approach based on fuzzy goal programing and quality function deployment for new product planning
Accepted Manuscript
Approach based on fuzzy goal programming and quality function deployment for new product planning Liang-Hsuan Chen , Wen-Chang Ko , Feng-Ting Yeh PII: DOI: Reference:
S0377-2217(16)30862-1 10.1016/j.ejor.2016.10.028 EOR 14051
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
5 August 2015 3 August 2016 15 October 2016
Please cite this article as: Liang-Hsuan Chen , Wen-Chang Ko , Feng-Ting Yeh , Approach based on fuzzy goal programming and quality function deployment for new product planning, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.10.028
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Highlights We proposed a new product planning (NPP) model to maximize customer satisfaction. Fuzzy goal programming and quality function deployment are used to construct this NPP model.
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Mathematical programming method is used to determine each customer requirement’s satisfaction expression. Experimental design and fuzzy sets are employed to collect the input-output data set in customer satisfaction model.
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We used Herzberg’s two-factor theory to enhance the total customer satisfaction.
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Approach based on fuzzy goal programming and quality function deployment for new product planning
a
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Liang-Hsuan Chena, Wen-Chang Ko*b, and Feng-Ting Yeha
Department of Industrial and Information Management, National Cheng Kung University, Tainan 710, Taiwan, R. O. C.
Department of Information Management, Kun Shan University, Tainan 710, Taiwan, R. O. C.
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b
* Corresponding author: Tel.: +886-6-2051630
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E-mail: [email protected]
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Fax: +886-6-2050587
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Approach based on fuzzy goal programming and quality function deployment for new product planning
Abstract Quality function deployment (QFD) is a useful planning tool for facilitating new product
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planning (NPP) to maximize customer satisfaction. Although customer satisfaction is an important goal in NPP, other goals must also be taken into account. The evaluation of QFD involves vagueness and imprecision and thus fuzzy approaches have been applied. This study considers the
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satisfaction of each customer requirement (CR) as the response variable and the fulfillment levels of the corresponding design requirements (DRs) as the explanatory variables. Each CR’s satisfaction expression is formulated using the mathematical programming method. Experimental design and fuzzy sets are employed to collect the input-output data set based on the evaluated
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relationships between CRs and DRs and the correlations among DRs in QFD processes based on the QFD team’s ability, experience, and knowledge. Considering three objectives, namely the
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maximum customer satisfaction, minimum incremental cost and minimum technical difficulty of
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DRs in NPP, an additive fuzzy goal programming model is proposed to obtain the optimal satisfaction under the preemptive priority structure of all goals. Furthermore, considering that
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customer satisfaction complies with Herzberg’s two-factor theory, this study incorporates the
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concept of motivation and hygiene factors into the model, to modify the fulfillment levels of DRs to enhance customer satisfaction. A numerical example is used to demonstrate the applicability of the proposed model. Keywords: Fuzzy goal programming (FGP), Quality function deployment (QFD), New product planning (NPP), Mathematical programming (MP), Two-factor theory
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ACCEPTED MANUSCRIPT 1. Introduction Faced with short-life cycles and dynamic competition in the global market, business units can efficiently develop new products that will be preferred by customers through the processes of new product development (NPD). New product planning (NPP) is the first stage of NPD [1]. Quality function deployment (QFD) is an efficient and useful technique for NPD to ensure that NPP meets
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customers’ expectations in terms of maximizing customer satisfaction [1-5]. Customer satisfaction is viewed as a vital competitive advantage because it directly impacts customer loyalty and market share [6-9]. To apply QFD to NPP, the systematic tool called house of quality (HOQ), shown in Fig.1, is usually employed. The QFD process starts with customer requirement (CR) (the voice of
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the customer) identification. A QFD team is organized to collect CRs and determine their importance scores. Based on the experience and judgment of the QFD team, the identified CRs are translated into product design requirements (DRs) to meet customer expectations in NPP. Calculating the relationship degrees between CRs and DRs (i.e., DRs that affect CRs) and the
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correlations among DRs with the importance scores of CRs to determine the importance ratings of
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DRs is the major task in NPP by using HOQ. According to the DRs’ importance ratings, the QFD team arranges adequate resources for DRs to ensure that CRs can be met and that customer
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satisfaction can be maximized in NPP.
[Insert Figure 1 around here]
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NPP is also called the fuzzy front end of NPD, since it involves vagueness and imprecision.
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Due to the inherent fuzziness in NPP, applying fuzzy methods to QFD is very appropriate for practical applications. Traditionally, the importance score of each CR is set as a crisp value [1, 10, 11], although linguistic terms seem more adequate for evaluating a CR’s importance. Furthermore, the relationship degree to which a DR affects a CR is expressed on a numerical scale, such as 1-3-9 or 1-5-9, representing the linguistic expressions ―weak‖, ―moderate‖, and ―strong‖, respectively. Alternatively, linguistic terms can be used to subjectively assess the relationships between CRs and DRs, as well as the relationships among DRs. Studies have thus applied fuzzy approaches to QFD 4
ACCEPTED MANUSCRIPT processes for determining customer satisfaction in NPP. Customer satisfaction is often determined using linear or nonlinear programming approaches to aggregate each DR’s contribution to CRs via the importance ratings and fulfillment levels of DRs [12-20]. However, a decision-maker (DM) usually has an aspiration level for each CR’s satisfaction degree in NPP. This leads to the important problem of how to adjust the fulfillment levels of different DRs by allocating resources to achieve each CR’s satisfaction degree under resource limitations. To deal with this problem, the fuzzy
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relationship between each CR’s satisfaction degree and the fulfillment levels of the associated DRs should be formulated beforehand to attain the aspiration level of each CR’s satisfaction degree and to find the maximal customer satisfaction under resource limitations. In the existing studies, some
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researchers have applied fuzzy regression, which was originally proposed by Tanaka et al. [21], with fuzzy and/or real assessments to QFD processes to estimate the relationships between CRs and DRs and the correlations among DRs [22-26]. In these studies, the observed data of the response and explanatory variables were collected from the alternatives in the competition analysis. However,
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during NPP, collecting observed data from competitors is difficult because NPP is confidential
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information.
In contrast with the existing approaches, this study adopts experimental design and fuzzy set
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theory to collect the observed data based on the QFD team’s knowledge and experience, considering the issue of observed data and the inherent fuzziness in NPP. Using the collected data
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set, the fuzzy mathematical model of each CR’s satisfaction level is constructed based on the
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fulfillment levels of DRs and the interaction of the fulfillment levels among DRs. The use of the fuzzy mathematical programming method has the advantage that the membership function obtained from the established formulation of each CR’s satisfaction degree has the most coincidence with that from the observed data. The distances between their membership functions are defined as errors, which are minimized in the mathematical model-fitting process. More importantly, only a few fuzzy observations from experimental design are needed for constructing mathematical models. In addition, some researchers have added incremental cost and technical difficulty into their 5
ACCEPTED MANUSCRIPT proposed models based on the consideration of NPP effort [12-14, 16-20, 27, 28]. This present study applies the fuzzy goal programming (FGP) approach to incorporate three objectives into the models, namely the maximum total customer satisfaction, minimum incremental cost and minimum technical difficulty of DRs. Furthermore, considering the QFD team’s preferences, the preemptive priority structure [28-34] of all goals is considered in the models as the constraint condition to attain the optimal satisfaction achievement of all goals in terms of the fulfillment levels of DRs.
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Particularly, this study also applies Herzberg’s two-factor theory [35] to modify the fulfillment levels of DRs in the FGP model for further enhancing customer satisfaction for NPP. Two factors, motivation and hygiene, in the theory have been widely applied to improve job satisfaction and
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performance in an organization. The use of motivation factors will lead to positive satisfaction, while hygiene factors are employed to avoid dissatisfaction. Likewise, for each CR’s satisfaction, the associated DRs can be categorized as motivation or hygiene factors in the FGP model based on their fulfillment levels in order to further increase customer satisfaction.
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The rest of this paper is organized as follows. CR satisfaction models of QFD based on
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mathematical programming are introduced in Section 2. In Section 3, FGP models are proposed by considering the maximum customer satisfaction, minimum incremental cost, and minimum
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technical difficulty of DRs under the relevant constraints to achieve the optimal satisfaction level for NPP. The motivation and hygiene factors of Herzberg’s two-factor theory are applied to enhance
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customer satisfaction based on the FGP models. A semi-conductor packing case is presented to
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illustrate the proposed approaches in Section 4. The conclusions are given in Section 5.
2. CR satisfaction model of QFD When QFD is applied to NPP, the relationship Rij between CRi and DRj and the correlation rjJ between DRj and DRJ in HOQ are assessed using defined fuzzy numbers based on the QFD team’s ability, experience, and knowledge. Considering the inherent fuzziness in NPP, evaluations of Rij and rjJ are usually carried out in terms of linguistic terms, i.e., fuzzy sets, that are characterized by 6
ACCEPTED MANUSCRIPT membership functions, and are denoted as Rij and rjJ , respectively. A fuzzy number can fully and uniquely be represented by its -cuts, [0, 1] [36], which are represented as closed intervals of real numbers. In general, the calculations of fuzzy numbers can be performed efficiently in terms of
-cuts. For instance, the -cut of Rij at the level can be expressed by its lower and upper bounds [( Rij )L ,( Rij )U ]
,
which
are
defined
as
( Rij )L inf{x Rij ( x) } x
and
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as
( Rij )U sup{x Rij ( x) } , where Rij ( x) is the membership degree of x belonging to Rij and is x
a defined fuzzy number in a linguistic scale.
Considering the fuzzy nature of the CR satisfaction level in NPP, the fuzzy satisfaction level of
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CRs and the corresponding DRs’ fulfillment levels are defined as the response variable and explanatory variables, respectively, based on the fuzzy assessment of the relationships between CRs and DRs, and the correlation among DRs in HOQ. Mathematically, the satisfaction level of the ith
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CR can be expressed as: J
J
J
yi 0( i ) (j i ) Rij x (ji ) (jhi ) rjh x (ji ) xh( i ) , j h
(1)
j 1 h 1
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j 1
where the response variable yi is the output that represents the fuzzy satisfaction level of CRi, and
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the explanatory variables x (ji ) and xh(i ) are real-value inputs that represent the fulfillment levels of
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DRj and DRh corresponding to Rij and rjh in HOQ, respectively, where x (ji ) and xh(i ) [0,1] . According to the -cuts and the extension principle [37, 38], Eq. (1) can be defined by the lower
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and upper bounds of each level as:
yi
0( i ) (j i ) Rij x (ji ) (jhi ) rjh x (ji ) xh( i )
(2a)
yi
0( i ) (j i ) Rij
(2b)
L
U
J
j 1
J
j 1
J
L
U
J
j 1 h 1
L
x (ji ) (jhi ) rjh J
J
j 1 h 1
U
x (ji ) xh( i )
where j h , x (ji ) = 0 implies that DR j has a basic fulfillment level to meet CRi so that no more
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ACCEPTED MANUSCRIPT resources are needed. To perform the above model-fitting process, this study employs experimental design and fuzzy set theory to collect the input-output data set. An orthogonal array is adopted to collect adequate observation data of the response variable and explanatory variables in NPP. Based on Eqs. (2a) and (2b), the coefficients of ( i ) = ( 0(i ) , (j i ) , (jhi ) , jh) should be determined to build up the optimal models. For this purpose, a mathematical programming approach is used to
R
L
ij
using
x(ji ) and
yi
L
r
L
jh
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find the optimal estimators of coefficients. The input data are calculated with various levels using
x(ji ) xh(i ) , where j = 1, …, J, h = 1, …, J, j h . The output data are determined
with the same levels as those used for the input data set. The level is interpreted
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as the confidence degree [39, 40]. With the objective of minimizing the errors between the observed and estimated fuzzy responses, denoted as yi and yˆ i respectively, a mathematical programming model has to be established to determine the estimators of ˆ ( i ) =( ˆ0(i ) , ˆ (j i ) , ˆ (jhi ) , j h) in the
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following equations:
yˆi
L L ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i )
yˆi
U U ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i )
j 1
J
j 1
J
j 1 h 1
J
J
(3a)
J
j 1 h 1
(3b)
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U
J
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L
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The errors between yˆ i and yi are considered as the distances between their membership functions from a geometrical viewpoint [41]. As such, the fuzzy error between yˆ i and yi at the
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kth -level is defined as
1 ( yˆi )Lk ( yi )Lk ( yˆi )Uk ( yi )Uk , which is actually the average of the 2
distance between the lower bounds and that between the upper bounds of yˆ i and yi at the kth level. To determine the optimal expression for the estimated fuzzy response of each CR, a set of observation data, consisting of the fulfillment levels of the corresponding DRs and a fuzzy response, should be collected. For the ith CR, the kth observed and estimated fuzzy responses are denoted as
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ACCEPTED MANUSCRIPT yik and yˆ ik , respectively. Let ik be the average of distances between yˆ ik and yik from M
-levels for the CRi satisfaction, which is formulated as: 1 2M
ik
( yˆ M
m 1
( yik )Lm ( yˆik )Um ( yik )Um
L ik m
)
(4)
The estimators of ˆ ( i ) =( ˆ0(i ) , ˆ (j i ) , ˆ (jhi ) , j h) in Eqs. (3a) and (3b) for the estimated fuzzy
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response of the CRi are determined by minimizing the sum of distances, i.e., Min ik , k = 1, …, K, based on K observations. As mentioned before, an experimental design is adopted to collect the observation data. For CRi, the kth observation includes the fuzzy response yik and the fulfillment
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levels of the corresponding DRs, x (jki ) , j=1, …, J. It is noted that some x (jki ) will be zero (they do not have any contribution to CRi). A mathematical programming problem for the CRi satisfaction model is formulated as follows: K
Min ik
M
k 1
s.t.
L L (i ) ˆ0(i ) ˆ (j i ) Rij x(jki ) ˆ (jhi ) rjh x(jki ) xhk J
L m
J
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yˆik
m
j 1
J
m
j 1 h 1
U U U yˆik ˆ0(i ) ˆ (j i ) Rij x(jki ) ˆ (jhi ) rjh x(jki ) xhk(i ) J
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J
m
m
j 1
J
m
j 1 h 1
k = 1, …, K; m = 1, …, M
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(5)
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Model (5) is a constrained nonlinear problem. In general, a commercial software package, such as Lingo, can be used to solve it. Summing the CRs’ importance scores, the fuzzy customer satisfaction of QFD for NPP can be
expressed as:
Yˆ
L
I
k yˆ i 1
L
i
i
and
(6a)
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Yˆ
U
I
k yˆ i 1
U
i
(6b)
i
where ki is the importance score of CRi, i=1, …, I;
k
i
1 . The outcomes of Eqs. (6a) and (6b)
are set as the maximization objective for NPP. The FGP models used to achieve the optimal
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satisfaction of all goals in terms of the fulfillment levels of DRs are introduced in the next section.
3. Formulations
This section introduces two FGP models for NPP. The concept of -cuts and the extension principle are applied to transform the FGP models into conventional linear programming models to
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determine the fulfillment levels of DRs that achieve the maximum satisfaction level of all goals. Customer satisfaction complies with Herzberg’s two-factor theory [35], and thus a modified FGP model that takes motivation and hygiene factors into account is used to further enhance customer satisfaction.
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3.1 FGP models
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The FGP model considers three objectives, namely the maximum customer satisfaction, minimum incremental cost, and minimum technical difficulty of DRs to assist DMs in determining
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the fulfillment levels of DRs to obtain the maximum sum of the satisfaction of objectives. According to the CR satisfaction model, the objective functions for maximizing customer
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satisfaction can be defined as follows: (7)
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I J J J max ki ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i ) i 1 j 1 j 1 h 1
~ where x (ji ) represents the decision variable, x (ji ) [0, 1]. Let C j and T j be the fuzzy
incremental cost and technical difficulty of achieving the fulfillment level of DRj, respectively. The business competition threshold j and the maximum possible ability j of DRj are considered as constraints for NPP. Furthermore, since the maximum satisfaction degree of each CR is 100%,
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ACCEPTED MANUSCRIPT the FGP model should be subject to the constraint that the satisfaction level of each CR cannot be greater than 1. Therefore, the FGP model is formulated as follows: I J J J max ki ˆ0(i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh(i ) i 1 j 1 j 1 h 1 J
min C j x j j 1 J
min T j x j
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j 1
s. t. J
J
J
ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i ) 1, i j 1
0
j
j 1 h 1
xj
j
1, j
(8)
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To deal with the above FGP model, this study adopts Tiwari et al.’s idea [30] of using the additive model to aggregate the fuzzy goals. First, the aspiration levels for each goal are determined by the DM. Then the membership functions of the satisfaction degree of max-goal and min-goal can
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be formulated as follows, respectively [42, 43]:
L 0, if Gmax ( x) Gmax , L ( x) Gmax G L U max (x) maxU , if Gmax Gmax ( x) Gmax , L Gmax Gmax U 1 , if Gmax ( x) Gmax .
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ED
(9)
U 0, if Gmin ( x) Gmin , U Gmin ( x) G L U min (x) minU , if Gmin Gmin ( x) Gmin , L G G min min L 1, if Gmin ( x) Gmin ,
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(10)
where x denotes the variable vector. Gmax (x) and Gmin (x) represent the satisfaction levels of max-goal and min-goal at x, respectively. Corresponding to Eq. (8), Gmax (x) represents J J ˆ (i ) J ˆ (i ) (i ) k R x ˆ (jhi ) rjh x(ji ) xh(i ) , while Gmin (x) denotes both i 0 j ij j i 1 j 1 j 1 h 1 I
J
C j x j and j 1
J
T x j 1
j
j
.
In Eqs. (9) and (10), L and U respectively specify the lower and upper bounds of the aspiration of the goal. Moreover, the preemptive priority structure is considered in the model. If three fuzzy goals 11
ACCEPTED MANUSCRIPT are considered as that the customer satisfaction (C. S.) is more important m times than the incremental cost expenditure, the preemptive priority structure can be denoted as membership functions and represented as C. S. (x) m Cost (x) , m>1. Therefore, Eq. (8) can be expressed as an additive model with a preemptive priority structure of goals and relevant constraints. Let 1 (x) ,
2 (x) , and 3 (x) represent the membership functions of satisfaction degrees of the customer
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satisfaction, cost, and technical difficulty, respectively. In addition, let p (x) , p, be greater than a threshold to ensure that none of the goals is ignored during the calculation. Combining the approaches above with the -cuts and extension principle in fuzzy sets [36, 37], Eq. (8) can be
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transformed into conventional linear programming models to determine the fulfillment levels of DRs that achieve the maximum satisfaction level of all goals. The membership function of the objective value Z in the linear programming model can be defined as
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Z ( Z ) sup min R ( Rij ), r (rjh ), C (C j ), T (T j ), i, j, h Z p ( x)
jh
ij
j
j
p 1
(11)
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R , r ,C ,T
where R, r, C, and T are the element values of the fuzzy coefficients defined in their corresponding
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linguistic scale. Applying Eq. (11), Z ( Z ) can be determined using the membership levels of all
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fuzzy coefficients. Therefore, Eq. (8) can be simplified to conventional linear programming models and separated into two crisp sub-additive models to find the lower and upper bounds of membership
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functions of the satisfaction degree, shown as Eqs. (12a) and (12b), respectively: 3
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ZL max p ( x) p 1
s. t.
1 ( x)
I
J
i 1
j 1
J
j 1 h 1
U L GC. S. GC. S. J
2 ( x)
J
ki ( ˆ0(i ) ˆ (j i ) ( Rij )L x (ji ) ˆ (jhi ) (rjh )L x (ji ) xh( i ) ) GC.L S. U GCost (C j )L x j j 1
U L GCost GCost
,
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,
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3 ( x )
U L GT. D. (T j ) x j j 1
,
L GT. D.
U T. D.
G
J
J
J
(i ) ˆ0( i ) ˆ (j i ) ( Rij )L x (ji ) ˆ jh ( rjh )L x (ji ) xh( i ) 1, i, j 1
j 1 h 1
t ( x) mv ( x), m 1, t , v {1, 2, 3},
(12a)
t v
0
j
xj
1, j ,
j
3
ZU max p ( x) p 1
s. t. J
i 1
j 1
J
U C. S.
G J
2 ( x)
U GCost (C j )U x j j 1 U Cost
,
L GCost
G
J
U U GT. D. (T j ) x j j 1
U T. D. J
G
J
,
L GT. D.
J
j 1 h 1 L C. S.
G
,
M
3 ( x )
J
ki ( ˆ0(i ) ˆ (j i ) ( Rij )U x (ji ) ˆ (jhi ) (rjh )U x(ji ) xh( i ) ) GC.L S.
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1 ( x)
I
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p ( x) 1, p.
ˆ0(i ) ˆ (j i ) ( Rij )U x (ji ) ˆ jh(i ) (rjh )U x (ji ) xh(i ) 1, i, j 1
j 1 h 1
t (x) mv (x), m 1, t , v {1, 2, 3}, 0 j x j j 1, j ,
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p (x) 1, p.
(12b)
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t v
levels.
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The membership function of Z can then be constructed using various [ ZL , ZU ] from different
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3.2 FGP models with Herzberg’s two-factor theory This study employs Herzberg’s two-factor theory to increase customer satisfaction.
Considering the fulfillment levels of DRs from Eqs. (12a) and (12b), DRs with the lower fulfillment levels comply with the motivation factor in Herzberg’s two-factor theory [35]. This means that when a DR’s fulfillment level is increased, customer satisfaction will increase, but decreasing a DR’s fulfillment level will not have a great impact on customer satisfaction. In contrast, DRs with
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ACCEPTED MANUSCRIPT higher fulfillment level comply with the hygiene factor in the two-factor theory. Increasing a DR’s fulfillment level will not enhance customer satisfaction significantly, but decreasing it will impact customer satisfaction. Therefore, a modified FGP model is proposed based on Eqs. (12a) and (12b), and the concept of Herzberg’s two-factor theory to enhance customer satisfaction for NPP as follows: 3
p 1
s. t. I
1 ( x)
J
k ( ˆ
(i ) 0
i
i 1
J
J
L ˆ (j i ) ( Rij )L x qj ( i ) ˆ (jhi ) ( rjh )L x qj ( i ) xhq ( i ) ) GC. S. j 1
U C. S.
G
j 1 h 1 L C. S.
G
J
U GCost (C j )L x qj j 1 U Cost
,
L GCost
G
J
3 ( x )
U L q GT. D. (T j ) x j j 1
U T. D.
G
,
L GT. D.
J
J
,
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2 ( x)
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ZL max p ( x)
J
M
ˆ0( i ) ˆ (j i ) ( Rij )L x qj ( i ) ˆ (jhi ) ( rjh )L x qj ( i ) xhq ( i ) 1, i, j 1
j 1 h 1
t ( x) mv ( x), m 1, t , v {1, 2, 3}, t v j
x qj
j
1, j ,
(13a)
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p ( x) 1, p.
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0
3
ZU max p ( x) s. t.
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p 1
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1 ( x)
2 ( x)
I
J
i 1
j 1
J
J
ki ( ˆ0(i ) ˆ (j i ) ( Rij )U x qj ( i ) ˆ (jhi ) (rjh )U x qj ( i ) xhq ( i ) ) GC.U S. U C. S.
G
j 1 h 1 L C. S.
G
J
U GCost (C j )U x qj j 1 U Cost
G
,
L GCost
J
3 ( x )
U U q GT. D. (T j ) x j
J
j 1
U T. D.
G
,
L GT. D. J
J
ˆ0( i ) ˆ (j i ) ( Rij )U x qj ( i ) ˆ (jhi ) ( rjh )U x qj ( i ) xhq ( i ) 1, i, j 1
j 1 h 1
14
,
ACCEPTED MANUSCRIPT t ( x) mv ( x), m 1, t , v {1, 2, 3}, t v
0
j
x qj
j
1, j ,
(13b)
p (x) 1, p, where q =0.5 or 1 if the corresponding DR is viewed as a motivation or hygiene factor, respectively. Of note, in Eqs. (13a) and (13b), the fulfillment level of DRj, j=1, 2, …, J, that is a motivation factor will be increased, whereas that of DRj that is a hygiene factor will be unchanged to increase
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customer satisfaction in the model.
4. Illustration
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A semiconductor packing case is used to illustrate the feasibility and applicability of the proposed models. In this section, the proposed model is applied to a turbo thermal ball grid array (T2-BGA). T2-BGA is a member of the plastic ball grid array (PBGA) family. Since a copper heat slug is lodged in the molding compound of PBGA, T2-BGA has good heat dissipation and electrical
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characteristics. However, T2-BGA still has some issues since its structure is more complex than that of PBGA [44]. The cross-sections of PBGA and T2-BGA are shown in Fig. 2.
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[Insert Figure 2 around here]
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According to the characteristics of T2-BGA, the QFD team collects five CRs, namely ―package profile‖ (CR1), ―thermal performance‖ (CR2), ―electrical performance‖ (CR3), ―reliability‖ (CR4),
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and ―co-planarity‖ (CR5). Considering these CRs, the QFD team develops the corresponding DRs using new product feasibility studies. Five DRs are determined, namely ―heat slug exposed area‖
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(DR1), ―heat slug attached material‖ (DR2), ―height of heat slug‖ (DR3), ―copper pattern‖ (DR4), and ―molding flow‖ (DR5). An HOQ for the CRs and DRs above is determined, as shown in Fig. 3. The fuzzy relationships and correlations are denoted as ―very weak (VW)‖, ―weak (W)‖, ―moderate (M)‖, ―high (H)‖, or ―very high (VH)‖, which are converted into triangular fuzzy numbers as the three-element sets (0, 0, .2), (0, 0.2, 0.4), (0.3, 0.5, 0.7), (0.6, 0.8, 1), and (0.8, 1, 1), respectively. The middle value in the three-element set represents the most likely one to have a membership
15
ACCEPTED MANUSCRIPT degree equivalent to 1, such as ( M ) (0.5) 1 . The membership functions are shown in Fig. 4. For instance, for Rij = ―M‖, the membership function is defined as: ( Rij 0.3) , 0.3 R1,i 0.5 (0.5 0.3) Rij ( Rij ) (0.7 Rij ) , 0.5 R 0.7 1,i (0.7 0.5)
CR IP T
[Insert Figures 3 and 4 around here]
To obtain each CR satisfaction degree model using Eqs. (2a) and (2b), the lower and upper bounds of each -cut of Rij and rj should be determined beforehand. The lower and upper
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bounds of each -cut can be represented as functions of . For example, the lower and upper bounds of each -cut of R1,2 can be expressed as [( R1,2 )L ,( R1,2 )U ] = [0.3 0.2 , 0 .7 0.2 ] . Let
e denote the eth level, [0, 1], where e 1 , e e E , and e e1 1 E , e {0,1, , E} .
M
This means that the distance between two arbitrarily adjacent levels is equal in [0, 1]. 4.1 Fuzzy customer satisfaction model of CR
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To collect the input-output data set, the CR satisfaction degree, in the fuzzy sense, and the
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corresponding DRs’ fulfillment levels are defined as the response variable and explanatory variables, respectively. The fuzzy satisfaction degrees are denoted as ―not satisfied (NS)‖, ―slightly satisfied
CE
(SS)‖, ―medium satisfied (MS)‖, ―satisfied (S)‖ or ―very satisfied (VS)‖, which are converted to triangular fuzzy numbers as the 3-element sets (0, 0, 0.3), (0, 0.25, 0.5), (0.3, 0.5, 0.7), (0.5, 0.75,
AC
1.0), and (0.7, 1.0, 1.0), respectively. Their membership functions are shown in Fig. 5. According to the HOQ of T2-BGA in Fig. 3, the QFD team employs experimental design to construct the initial input-output data sets by considering various portfolios between each CR’s fuzzy satisfaction degree and the corresponding DRs’ fulfillment levels based on the QFD team’s experience and domain knowledge. The initial input-output data sets for the CRs are listed in Tables 1 to 5. Each sample in these tables can be considered a different confidence level (i.e. -level) of the fuzzy
16
ACCEPTED MANUSCRIPT satisfaction degree. In this study, three levels are adopted, {0.2, 0.8, 1}, for each sample such that five satisfaction degrees (output data) are collected from each sample in Tables 1 to 5. The data of DR fulfillment levels in Tables 1 to 5 have to be combined with their fuzzy relationships and correlations (shown in Fig. 3) with the same confidence levels to determine the corresponding input data for formulating each CR’s satisfaction model. Employing Eqs. (4) and (5), the vectors of the
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estimators of Eq. (3), ˆ ( i ) , i=1, …, 5, can be obtained. The predicted fuzzy satisfaction degree expression of each CR can be expressed as:
yˆ1 0.43R12 x2 0.381R13 x3 2.021R15 x5 0.118r23 x2 x3 0.542r25 x2 x5
(14-1)
yˆ 2 1.333R21 x1 2.611R24 x4
(14-2)
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yˆ3 1.25R31 x1 1.318R34 x4
(14-3) (14-4)
yˆ5 0.24 2.5R54 x4
(14-5)
M
yˆ4 0.037 0.599R42 x2 2.132R45 x5 0r25 x2 x5
The predicted fuzzy satisfaction degree expression for each CR (Eqs. (14-1) to (14-5)) can be
ED
expressed by the lower and upper bounds of the level, as done in Eqs. (3a) and (3b). Suppose that the importance score of each DR is determined by the QFD team as {0.14, 0.24, 0.17, 0.32, 0.13}.
CE
then be obtained.
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Using Eqs. (6a) and (6b), the lower and upper bounds of the fuzzy customer satisfaction model can
[Insert Figure 5 around here]
[Insert Tables 1 and 5 around here]
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4.2 FGP models
Referring to Eq. (8), the QFD team determines the fuzzy incremental cost and technical
difficulty for each DR, which are listed in Table 6. To transform the FGP model into a linear programming model, the lower and upper bounds of the aspiration levels of the three goals are set as shown in Fig. 6. Based on Table 6 and Fig. 6, the membership functions of satisfaction degree of customer satisfaction, incremental cost, and technical difficulty can be formulated using Eqs. (9)
17
ACCEPTED MANUSCRIPT and (10) as follows:
C. S. (x) [(0.043 0.1065 x1 0.1834 x2 0.0267 x3 0.3024 x4 0.193x5 0.0132 x2 x3 0.0152 x2 x5 ) 0.3] / 4
Cost (x) [1.8 (0.25 x1 0.75 x2 0.5 x3 0.5 x4 0.25 x5 )] / 0.8
(15) (16)
T. D. (x) [2.0 (0.75 x1 0.25 x2 0.5 x3 0.25 x4 0.75 x5 )] /1.2 (17)
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Considering the preemptive priority structure of goals, suppose that the goal of the customer satisfaction is 1.05 times more important than those of incremental cost and technical difficulty. The threshold is set to 0.05 for each goal. The business competition threshold j and the maximum possible ability j of DRj , j = 1,…,5, are set as 0.2 and 0.95, respectively. According to the
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information above, the various lower and upper membership levels of fuzzy goals and the fulfillment levels of DRs can be obtained using Eqs. (12a) and (12b). Table 7 lists the outcomes of Eqs. (12a) and (12b) for the present case.
M
[Insert Figure 6 around here] [Insert Table 6 around here] [Insert Table 7 around here] To enhance the customer satisfaction degree, the outcomes of the fulfillment levels of DRs in
ED
Table 7 are defuzzified and categorized into motivation and hygiene factors. A DR is categorized as a motivation factor if its defuzzified fulfillment level is less than 0.7; otherwise, it is classified as a
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hygiene factor. In this case, the fuzzy mean (FM) method [45] is adopted to perform the
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defuzzification of xi , i = 1,…, 5, since it is simple and efficient. F
f 1 F
(18)
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d
f xf f f 1
where f and x f denote the membership degree and the representative numerical value of the fth output, respectively. In this study, d is the defuzzification value of interval values [ x Lf , x Uf ], i.e., the fth -cut of the fuzzy number. Applying Mabuchi’s idea [46], the representative value of the fth
-cut is defined as the average of the lower and upper bounds of the interval: x f 1 2 ( x Lf x Uf ) . 18
ACCEPTED MANUSCRIPT Based on the FM method and Mabuchi’s idea, the fulfillment levels of DR1, …, DR5 can be defuzzified as {0.2, 0.557, 0.2, 0.95, 0.708} based on an level set with {1, 0.95, 0.9}. Therefore, DR1, DR2, and DR3 are categorized as the motivation factors, and DR4 and DR5 are classified as hygiene factors. According to this DR categorization, the acceptable confidence degree must be greater than or equal to 0.5. The various lower and upper membership levels of fuzzy goals are
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updated and the modified fulfillment levels of DRs can be obtained using Eqs. (13a) and (13b). Table 8 lists the outcomes of Eqs. (13a) and (13b) for T2-BGA. In Table 8, the acceptable confidence degree produces infeasible regions at 0.5 in Eq. (13b) of the QFD processes. [Insert Table 8 around here]
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4.3 Discussion
In the present case, 90 observed data were collected using experimental design and fuzzy set theory based on the QFD team’s knowledge and experience. This approach is reasonable for NPP in a practical (fuzzy) environment. The formulations of each CR’s satisfaction degree are listed in Eqs
M
(14-1) to (14-5). They are useful for determining the models of total customer satisfaction degree in
ED
QFD processes.
For comparison, the customer satisfaction degree of models (13a) and (13b), as well as (12a)
PT
and (12b), were determined using Eqs. (6a) and (6b) based on the outcomes of the fulfillment levels of DRs in Tables 7 and 8. Models (13a) and (13b), as well as (12a) and (12b), include 24 constraints
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and 8 decision variables.
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Referring to Tables 7 and 8, the outcome of C.S. from models (13a) and (13b) is better than that from models (12a) and (12b) at all levels; on the other hand, the outcomes of Cost and
T. D. from models (13a) and (13b) are less than those from models (12a) and (12b) at all levels under the same constraints. The outcomes of C.S. , Cost and T. D. in Table 8 denote the weight assignments of the three goals. Figure 7 shows the customer satisfaction degree based on models (13a) and (13b), as well as (12a) and (12b), of the QFD processes for levels greater than or equal 19
ACCEPTED MANUSCRIPT to 0.9. The customer satisfaction degrees for models (13a) and (13b) (i.e., with two-factor theory idea, YTFT) are better than those for models (12a) and (12b) for NPP of T2-BGA. In this case, at = 1, the customer satisfaction degree for models (13a) or (13b) is enhanced by 14.1% compared to that for models (12a) or (12b) based on the outcomes of DR fulfillment levels in Tables 7 and 8. In Table 7 or 8, it is noted that maximizing the model may cause some decision variables at the
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lower bound to be greater than those at the upper bound for their corresponding objective functions. For example, in Tables 7, x5(L) is greater than x5(U) , at = 0.95 and x2(L) is greater than x2(U) at
= 0.9. However, the maximum membership degrees of all goals at the upper bound are greater than those at the lower bound for both proposed models. To avoid confusion, the optimal fulfillment
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levels of DRs determined for the lower and upper bounds of the membership degree of satisfaction degree are respectively denoted as (L) and (U) in the tables. Considering the fuzziness of NPP, the lower acceptable confidence levels may produce infeasible regions when the proposed models are
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solved. Once the infeasible regions are found, the DM has to determine a higher acceptable confidence degree to reduce uncertainty for obtaining adequate decision-making information for
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NPP. Models (5), (12a) and (12b), (13a) and (13b) were solved using Lingo 10 software in a Microsoft Windows 7 environment running on a laptop with a 2.1-GHz processor and 2 GB random
CE
calculation.
PT
access memory. The efficiency was satisfactory, with an almost negligible runtime for each
Although QFD was originally developed for product improvements or NPP, it has been
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successfully applied in various fields and industries [1-5, 47-50], as QFD frameworks are an effective approach for describing the relationships between ―what is needed‖ and ―how to do‖ in problems. As such, even though the proposed approaches are developed for NPP in this study, they can be easily applied to other fields or problems. The input-output observation data set was collected by carrying out an experiment design. To obtain legitimate observations, QFD team members should be capable of providing the corresponding fuzzy responses based on the
20
ACCEPTED MANUSCRIPT combinations of fulfillment levels of corresponding DRs. Inappropriate fuzzy responses will result in incorrect model-fittings for CRs and the subsequent decision for customer satisfaction.
5. Conclusion Determining the fulfillment levels of DRs to meet CRs to achieve optimal customer
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satisfaction is a major task in NPP. QFD is a systematic tool for NPP. This study considered three objectives, namely the maximum customer satisfaction, minimum incremental cost, and minimum technical difficulty, of DRs for NPP in a fuzzy environment. Applying the FGP approach and considering the preemptive priority of the various goals, each CR’s optimal satisfaction can be
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attained based on the overall goals in terms of the fulfillment levels of DRs. Each CR’s satisfaction level and the fulfillment levels of the corresponding DRs are defined as the response variable and explanatory variables, respectively. Each CR’s satisfaction expression is determined using the mathematical programming method. The observed data of the response and explanatory variables
M
are collected from the alternatives in the competition analysis. Experimental design and fuzzy sets
ED
are employed to collect the input-output data set based on the evaluated relationships between CRs and DRs and the correlations among DRs in QFD processes using the QFD team’s knowledge and
PT
experience. In addition, Herzberg’s two-factor theory is incorporated into the models of customer satisfaction. Updated models were proposed to enhance customer satisfaction for NPP.
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The proposed models were illustrated with a numerical example to demonstrate their
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applicability in practice. The resulting ranges of satisfaction degrees and the possible ranges of fulfillment levels of DRs in both models can provide the QFD team with useful information for new product design. The model that considers two-factor theory produced better outcomes of customer satisfaction of all goals. In future research, multiple segmentations of the market and the integration of the more ―what‖-―how‖ information of QFD activities can be considered for developing more sophisticated models.
21
ACCEPTED MANUSCRIPT Acknowledgement This research was funded by Contract NSC 98-2410-H-006-042-MY3 and in part by NSC 101-2410-H-168-003 from the Ministry of Science and Technology, Republic of China.
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Captions for figures and tables Figure 1. House of quality. Figure 2. Cross-section diagrams of PBGA and T2-BGA. Figure 3. HOQ for T2BGA. Figure 4. Membership functions of linguistic terms for relationships and correlations in HOQ. Figure 5. Membership functions of linguistic terms for ranking of CR satisfaction levels. 26
ACCEPTED MANUSCRIPT Figure 6. Lower and upper bounds of aspiration levels of three goals. Figure 7. Membership functions of customer satisfaction degree based on models (12) and (13). Table 1. Initial input-output data sets of CR1 and DRs. Table 2. Initial input-output data sets of CR2 and DRs. Table 3. Initial input-output data sets of CR3 and DRs.
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Table 4 Initial input-output data sets of CR4 and DRs. Table 5 Initial input-output data sets of CR5 and DR4.
Table 6. Fuzzy incremental cost and technical difficulty for each DR
Table 7 Outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various levels.
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Table 8 Updated outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various levels.
rjJ
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correlation among DRs
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CR1 k1 . . . . . .
Importance scores of CRs
DR1 . . . DRj . . . DRJ
ki . . .
CRI
kI
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CRi . . .
Relationship between CRi and DRj Rij
Importance ratings of DRs W1 . . . Wj . . . WJ
Figure 1. House of quality.
27
ACCEPTED MANUSCRIPT Chip
Gold wire
Heat slug
Solder ball
Molding compound
T2 - BGA
PBGA
Substrate
W H DR1 DR2 DR3 DR4 DR5 CR1 CR2
W
CR3
W
W
M
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M
W M
H
CR4
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Figure 2.Cross-section diagrams of PBGA and T2-BGA.
W
W
M
CR5
1 VW W
M
AC
CE
PT
Membership degree
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Figure 3. HOQ for T2BGA.
0
H
.2 .3 .4 .5 .6 .7 .8
VH
1
Linguistic terms
Figure 4. Membership functions of linguistic terms for relationships and correlations in HOQ.
28
ACCEPTED MANUSCRIPT NS
MS
SS
S
VS
Membership degree
1
0 1
2
3
4
5
6
7
8
9
10
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0
0
Membership degree
0
0.7 GC .S . (x)
1
1.8 GCost (x)
1
0
M
0.3
1
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1
Membership degree
Membership degree
Figure 5. Membership functions of linguistic terms for ranking of CR satisfaction levels.
0.8
2.0 GT . D. (x)
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Figure 6. Lower and upper bounds of aspiration levels of three goals.
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CE
Membership degree
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1
Yˆ
Yˆ TFT
0.99
0.98 0.58
0.6 0.62 0.64 0.66 0.68 0.7 Customer satisfaction degree
Figure 7. Membership functions of customer satisfaction degree based on models (12) and (13).
29
ACCEPTED MANUSCRIPT Table 1. Initial input-output data sets of CR1 and DRs. Satisfaction level of CR1
Fulfillment levels of DR2, DR3, and DR5 Sample x3
x5
x2 x 3
x2 x 5
y1
1 2 3 4 5 6 7
0.00 0.00 0.00 0.50 0.50 0.50 1.00
0.00 0.50 1.00 0.00 0.50 1.00 0.00
0.00 0.50 1.00 0.00 0.50 1.00 0.50
0.00 0.00 0.00 0.00 0.25 0.50 0.00
0.00 0.00 0.00 0.00 0.25 0.50 0.50
NS MS S NS MS VS MS
8 9 10 11 12 13 14 15 16 17
1.00 1.00 0.00 0.00 0.00 0.50 0.50 0.50 1.00 1.00
0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50
1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00
0.50 1.00 0.00 0.00 0.00 0.00 0.25 0.50 0.00 0.50
1.00 0.00 0.00 0.00 0.00 0.25 0.50 0.00 0.50 1.00
18
1.00
1.00
0.00
1.00
0.00
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x2
MS
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M
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VS MS NS MS NS SS S SS MS VS
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Table 2. Initial input-output data sets of CR2 and DRs. Fulfillment levels of DR1 and DR4
Satisfaction level of CR2
Sample
x4
y2
1 2
0 0
0 0.5
NS SS
3 4 5 6 7 8 9 10 11
0 0.5 0.5 0.5 1 1 1 0 0
1 0 0.5 1 0 0.5 1 0 0.5
MS NS MS S SS S VS NS SS
AC
CE
x1
30
ACCEPTED MANUSCRIPT 12 13
0 0.5
1 0
MS NS
14 15 16 17 18
0.5 0.5 1 1 1
0.5 1 0 0.5 1
MS S SS S VS
Fulfillment levels of DR1 and DR4 Sample x4
1 2
0 0
0 0.5
3 4 5 6 7 8
0 0.5 0.5 0.5 1 1
9 10 11 12 13 14 15 16 17 18
1 0 0 0 0.5 0.5 0.5 1 1 1
y3
NS SS
1 0 0.5 1 0 0.5
MS NS MS VS SS S
1 0 0.5 1 0 0.5 1 0 0.5 1
VS NS SS MS NS MS VS SS S VS
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ED PT
CE AC
Satisfaction level of CR3
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x1
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Table 3. Initial input-output data sets of CR3 and DRs.
Table 4. Initial input-output data sets of CR4 and DRs. Fulfillment levels of DR2 and DR5
Satisfaction level of CR4
x2
x5
x2 x 5
y4
0.00 0.50
0.00 0.50
0.00 0.25
NS MS
Sample 1 2
31
ACCEPTED MANUSCRIPT 1.00 0.00
1.00 0.00
1.00 0.00
VS NS
5 6 7 8 9 10 11 12 13
0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00
0.50 1.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50
0.25 1.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00
MS VS SS S MS SS S MS SS
14 15 16 17 18
0.50 1.00 0.00 0.50 1.00
1.00 0.00 0.50 1.00 0.00
0.50 0.00 0.00 0.50 0.00
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3 4
S MS SS S MS
Table 5. Initial input-output data sets of CR5 and DR4. Fulfillment levels of DR4
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CE 10 11 12 13 14 15 16 17 18
Satisfaction level of CR4
x4
y5
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
NS MS VS NS MS S NS MS S
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
NS MS S NS MS VS NS MS S
ED
PT
1 2 3 4 5 6 7 8 9
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Sample
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ACCEPTED MANUSCRIPT Table 6. Fuzzy incremental cost and technical difficulty for each DR DRj
Cj
-cut of C j
DR1
(0, 0.25, 0.5)
[0.25, 0.5-0.25]
-cut of T j
Tj
(0.5, 0.75, 1) [0.5+0.25, 1-0.25]
DR2 (0.5, 0.75, 1.) [0.5+0.25, 1-0.25] (0, 0.25, 0.5)
[0.25, 0.5-0.25]
DR3 (0.3, 0.5, 0.7) [0.3+0.2, 0.7-0.3] (0.3, 0.5, 0.7) [0.3+0.1, 0.7-0.3] DR4 (0.3, 0.5, 0.7) [0.3+0.2, 0.7-0.3] (0, 0.25, 0.5)
[0.25, 0.5-0.25]
[0.25, 0.5-0.25]
(0.5, 0.75, 1) [0.5+0.25, 1-0.25]
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DR5
(0, 0.25, 0.5)
Table 7 Outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various levels.
L U L U L U C.S. C.S. Cost Cost T.D. T.D. x1(L) x1(U) x2(L)
0.95 0.709 0.794 0.676 0.756 0.676 0.756 0.2 0.9 0.666 0.889 0.635 0.847 0.635 0.692 0.2
x3(L)
0.2 0.576 0.576 0.2
x3(U)
x4(L)
x4(U)
x5(L)
x5(U)
0.2 0.95 0.95 0.678 0.678
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1 0.752 0.752 0.717 0.717 0.717 0.717 0.2
x2(U)
0.2 0.572 0.58 0.2
0.2 0.95 0.95 0.701 0.655
0.2 0.568 0.464 0.2
0.2 0.95 0.95 0.725 0.824
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Table 8 Updated outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various
1
L U L U L U C.S. C.S. Cost Cost T.D. T.D.
x1L
x1U
x2L
x2U
x3L
x3U
x4L
x4U
x5L
x5U
0.964 0.964 0.19 0.19 0.209 0.209 0.447 0.447 0.447 0.447 0.447 0.447 0.95 0.95 0.678 0.678
PT
ED
levels.
- 0.145 - 0.186
-
0.447
-
AC
0.95 0.893
CE
0.98 0.936 0.9930.1720.208 0.2 0.218 0.447 0.447 0.447 0.447 0.447 0.447 0.95 0.95 0.669 0.687
33
0.447
-
0.447
-
0.95
-
0.655
-
Approach based on fuzzy goal programming and quality function deployment for new product planning Liang-Hsuan Chen , Wen-Chang Ko , Feng-Ting Yeh PII: DOI: Reference:
S0377-2217(16)30862-1 10.1016/j.ejor.2016.10.028 EOR 14051
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
5 August 2015 3 August 2016 15 October 2016
Please cite this article as: Liang-Hsuan Chen , Wen-Chang Ko , Feng-Ting Yeh , Approach based on fuzzy goal programming and quality function deployment for new product planning, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.10.028
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights We proposed a new product planning (NPP) model to maximize customer satisfaction. Fuzzy goal programming and quality function deployment are used to construct this NPP model.
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Mathematical programming method is used to determine each customer requirement’s satisfaction expression. Experimental design and fuzzy sets are employed to collect the input-output data set in customer satisfaction model.
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We used Herzberg’s two-factor theory to enhance the total customer satisfaction.
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Approach based on fuzzy goal programming and quality function deployment for new product planning
a
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Liang-Hsuan Chena, Wen-Chang Ko*b, and Feng-Ting Yeha
Department of Industrial and Information Management, National Cheng Kung University, Tainan 710, Taiwan, R. O. C.
Department of Information Management, Kun Shan University, Tainan 710, Taiwan, R. O. C.
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b
* Corresponding author: Tel.: +886-6-2051630
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E-mail: [email protected]
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Fax: +886-6-2050587
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Approach based on fuzzy goal programming and quality function deployment for new product planning
Abstract Quality function deployment (QFD) is a useful planning tool for facilitating new product
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planning (NPP) to maximize customer satisfaction. Although customer satisfaction is an important goal in NPP, other goals must also be taken into account. The evaluation of QFD involves vagueness and imprecision and thus fuzzy approaches have been applied. This study considers the
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satisfaction of each customer requirement (CR) as the response variable and the fulfillment levels of the corresponding design requirements (DRs) as the explanatory variables. Each CR’s satisfaction expression is formulated using the mathematical programming method. Experimental design and fuzzy sets are employed to collect the input-output data set based on the evaluated
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relationships between CRs and DRs and the correlations among DRs in QFD processes based on the QFD team’s ability, experience, and knowledge. Considering three objectives, namely the
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maximum customer satisfaction, minimum incremental cost and minimum technical difficulty of
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DRs in NPP, an additive fuzzy goal programming model is proposed to obtain the optimal satisfaction under the preemptive priority structure of all goals. Furthermore, considering that
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customer satisfaction complies with Herzberg’s two-factor theory, this study incorporates the
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concept of motivation and hygiene factors into the model, to modify the fulfillment levels of DRs to enhance customer satisfaction. A numerical example is used to demonstrate the applicability of the proposed model. Keywords: Fuzzy goal programming (FGP), Quality function deployment (QFD), New product planning (NPP), Mathematical programming (MP), Two-factor theory
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ACCEPTED MANUSCRIPT 1. Introduction Faced with short-life cycles and dynamic competition in the global market, business units can efficiently develop new products that will be preferred by customers through the processes of new product development (NPD). New product planning (NPP) is the first stage of NPD [1]. Quality function deployment (QFD) is an efficient and useful technique for NPD to ensure that NPP meets
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customers’ expectations in terms of maximizing customer satisfaction [1-5]. Customer satisfaction is viewed as a vital competitive advantage because it directly impacts customer loyalty and market share [6-9]. To apply QFD to NPP, the systematic tool called house of quality (HOQ), shown in Fig.1, is usually employed. The QFD process starts with customer requirement (CR) (the voice of
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the customer) identification. A QFD team is organized to collect CRs and determine their importance scores. Based on the experience and judgment of the QFD team, the identified CRs are translated into product design requirements (DRs) to meet customer expectations in NPP. Calculating the relationship degrees between CRs and DRs (i.e., DRs that affect CRs) and the
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correlations among DRs with the importance scores of CRs to determine the importance ratings of
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DRs is the major task in NPP by using HOQ. According to the DRs’ importance ratings, the QFD team arranges adequate resources for DRs to ensure that CRs can be met and that customer
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satisfaction can be maximized in NPP.
[Insert Figure 1 around here]
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NPP is also called the fuzzy front end of NPD, since it involves vagueness and imprecision.
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Due to the inherent fuzziness in NPP, applying fuzzy methods to QFD is very appropriate for practical applications. Traditionally, the importance score of each CR is set as a crisp value [1, 10, 11], although linguistic terms seem more adequate for evaluating a CR’s importance. Furthermore, the relationship degree to which a DR affects a CR is expressed on a numerical scale, such as 1-3-9 or 1-5-9, representing the linguistic expressions ―weak‖, ―moderate‖, and ―strong‖, respectively. Alternatively, linguistic terms can be used to subjectively assess the relationships between CRs and DRs, as well as the relationships among DRs. Studies have thus applied fuzzy approaches to QFD 4
ACCEPTED MANUSCRIPT processes for determining customer satisfaction in NPP. Customer satisfaction is often determined using linear or nonlinear programming approaches to aggregate each DR’s contribution to CRs via the importance ratings and fulfillment levels of DRs [12-20]. However, a decision-maker (DM) usually has an aspiration level for each CR’s satisfaction degree in NPP. This leads to the important problem of how to adjust the fulfillment levels of different DRs by allocating resources to achieve each CR’s satisfaction degree under resource limitations. To deal with this problem, the fuzzy
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relationship between each CR’s satisfaction degree and the fulfillment levels of the associated DRs should be formulated beforehand to attain the aspiration level of each CR’s satisfaction degree and to find the maximal customer satisfaction under resource limitations. In the existing studies, some
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researchers have applied fuzzy regression, which was originally proposed by Tanaka et al. [21], with fuzzy and/or real assessments to QFD processes to estimate the relationships between CRs and DRs and the correlations among DRs [22-26]. In these studies, the observed data of the response and explanatory variables were collected from the alternatives in the competition analysis. However,
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during NPP, collecting observed data from competitors is difficult because NPP is confidential
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information.
In contrast with the existing approaches, this study adopts experimental design and fuzzy set
PT
theory to collect the observed data based on the QFD team’s knowledge and experience, considering the issue of observed data and the inherent fuzziness in NPP. Using the collected data
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set, the fuzzy mathematical model of each CR’s satisfaction level is constructed based on the
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fulfillment levels of DRs and the interaction of the fulfillment levels among DRs. The use of the fuzzy mathematical programming method has the advantage that the membership function obtained from the established formulation of each CR’s satisfaction degree has the most coincidence with that from the observed data. The distances between their membership functions are defined as errors, which are minimized in the mathematical model-fitting process. More importantly, only a few fuzzy observations from experimental design are needed for constructing mathematical models. In addition, some researchers have added incremental cost and technical difficulty into their 5
ACCEPTED MANUSCRIPT proposed models based on the consideration of NPP effort [12-14, 16-20, 27, 28]. This present study applies the fuzzy goal programming (FGP) approach to incorporate three objectives into the models, namely the maximum total customer satisfaction, minimum incremental cost and minimum technical difficulty of DRs. Furthermore, considering the QFD team’s preferences, the preemptive priority structure [28-34] of all goals is considered in the models as the constraint condition to attain the optimal satisfaction achievement of all goals in terms of the fulfillment levels of DRs.
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Particularly, this study also applies Herzberg’s two-factor theory [35] to modify the fulfillment levels of DRs in the FGP model for further enhancing customer satisfaction for NPP. Two factors, motivation and hygiene, in the theory have been widely applied to improve job satisfaction and
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performance in an organization. The use of motivation factors will lead to positive satisfaction, while hygiene factors are employed to avoid dissatisfaction. Likewise, for each CR’s satisfaction, the associated DRs can be categorized as motivation or hygiene factors in the FGP model based on their fulfillment levels in order to further increase customer satisfaction.
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The rest of this paper is organized as follows. CR satisfaction models of QFD based on
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mathematical programming are introduced in Section 2. In Section 3, FGP models are proposed by considering the maximum customer satisfaction, minimum incremental cost, and minimum
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technical difficulty of DRs under the relevant constraints to achieve the optimal satisfaction level for NPP. The motivation and hygiene factors of Herzberg’s two-factor theory are applied to enhance
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customer satisfaction based on the FGP models. A semi-conductor packing case is presented to
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illustrate the proposed approaches in Section 4. The conclusions are given in Section 5.
2. CR satisfaction model of QFD When QFD is applied to NPP, the relationship Rij between CRi and DRj and the correlation rjJ between DRj and DRJ in HOQ are assessed using defined fuzzy numbers based on the QFD team’s ability, experience, and knowledge. Considering the inherent fuzziness in NPP, evaluations of Rij and rjJ are usually carried out in terms of linguistic terms, i.e., fuzzy sets, that are characterized by 6
ACCEPTED MANUSCRIPT membership functions, and are denoted as Rij and rjJ , respectively. A fuzzy number can fully and uniquely be represented by its -cuts, [0, 1] [36], which are represented as closed intervals of real numbers. In general, the calculations of fuzzy numbers can be performed efficiently in terms of
-cuts. For instance, the -cut of Rij at the level can be expressed by its lower and upper bounds [( Rij )L ,( Rij )U ]
,
which
are
defined
as
( Rij )L inf{x Rij ( x) } x
and
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as
( Rij )U sup{x Rij ( x) } , where Rij ( x) is the membership degree of x belonging to Rij and is x
a defined fuzzy number in a linguistic scale.
Considering the fuzzy nature of the CR satisfaction level in NPP, the fuzzy satisfaction level of
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CRs and the corresponding DRs’ fulfillment levels are defined as the response variable and explanatory variables, respectively, based on the fuzzy assessment of the relationships between CRs and DRs, and the correlation among DRs in HOQ. Mathematically, the satisfaction level of the ith
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CR can be expressed as: J
J
J
yi 0( i ) (j i ) Rij x (ji ) (jhi ) rjh x (ji ) xh( i ) , j h
(1)
j 1 h 1
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j 1
where the response variable yi is the output that represents the fuzzy satisfaction level of CRi, and
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the explanatory variables x (ji ) and xh(i ) are real-value inputs that represent the fulfillment levels of
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DRj and DRh corresponding to Rij and rjh in HOQ, respectively, where x (ji ) and xh(i ) [0,1] . According to the -cuts and the extension principle [37, 38], Eq. (1) can be defined by the lower
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and upper bounds of each level as:
yi
0( i ) (j i ) Rij x (ji ) (jhi ) rjh x (ji ) xh( i )
(2a)
yi
0( i ) (j i ) Rij
(2b)
L
U
J
j 1
J
j 1
J
L
U
J
j 1 h 1
L
x (ji ) (jhi ) rjh J
J
j 1 h 1
U
x (ji ) xh( i )
where j h , x (ji ) = 0 implies that DR j has a basic fulfillment level to meet CRi so that no more
7
ACCEPTED MANUSCRIPT resources are needed. To perform the above model-fitting process, this study employs experimental design and fuzzy set theory to collect the input-output data set. An orthogonal array is adopted to collect adequate observation data of the response variable and explanatory variables in NPP. Based on Eqs. (2a) and (2b), the coefficients of ( i ) = ( 0(i ) , (j i ) , (jhi ) , jh) should be determined to build up the optimal models. For this purpose, a mathematical programming approach is used to
R
L
ij
using
x(ji ) and
yi
L
r
L
jh
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find the optimal estimators of coefficients. The input data are calculated with various levels using
x(ji ) xh(i ) , where j = 1, …, J, h = 1, …, J, j h . The output data are determined
with the same levels as those used for the input data set. The level is interpreted
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as the confidence degree [39, 40]. With the objective of minimizing the errors between the observed and estimated fuzzy responses, denoted as yi and yˆ i respectively, a mathematical programming model has to be established to determine the estimators of ˆ ( i ) =( ˆ0(i ) , ˆ (j i ) , ˆ (jhi ) , j h) in the
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following equations:
yˆi
L L ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i )
yˆi
U U ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i )
j 1
J
j 1
J
j 1 h 1
J
J
(3a)
J
j 1 h 1
(3b)
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U
J
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L
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The errors between yˆ i and yi are considered as the distances between their membership functions from a geometrical viewpoint [41]. As such, the fuzzy error between yˆ i and yi at the
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kth -level is defined as
1 ( yˆi )Lk ( yi )Lk ( yˆi )Uk ( yi )Uk , which is actually the average of the 2
distance between the lower bounds and that between the upper bounds of yˆ i and yi at the kth level. To determine the optimal expression for the estimated fuzzy response of each CR, a set of observation data, consisting of the fulfillment levels of the corresponding DRs and a fuzzy response, should be collected. For the ith CR, the kth observed and estimated fuzzy responses are denoted as
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ACCEPTED MANUSCRIPT yik and yˆ ik , respectively. Let ik be the average of distances between yˆ ik and yik from M
-levels for the CRi satisfaction, which is formulated as: 1 2M
ik
( yˆ M
m 1
( yik )Lm ( yˆik )Um ( yik )Um
L ik m
)
(4)
The estimators of ˆ ( i ) =( ˆ0(i ) , ˆ (j i ) , ˆ (jhi ) , j h) in Eqs. (3a) and (3b) for the estimated fuzzy
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response of the CRi are determined by minimizing the sum of distances, i.e., Min ik , k = 1, …, K, based on K observations. As mentioned before, an experimental design is adopted to collect the observation data. For CRi, the kth observation includes the fuzzy response yik and the fulfillment
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levels of the corresponding DRs, x (jki ) , j=1, …, J. It is noted that some x (jki ) will be zero (they do not have any contribution to CRi). A mathematical programming problem for the CRi satisfaction model is formulated as follows: K
Min ik
M
k 1
s.t.
L L (i ) ˆ0(i ) ˆ (j i ) Rij x(jki ) ˆ (jhi ) rjh x(jki ) xhk J
L m
J
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yˆik
m
j 1
J
m
j 1 h 1
U U U yˆik ˆ0(i ) ˆ (j i ) Rij x(jki ) ˆ (jhi ) rjh x(jki ) xhk(i ) J
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J
m
m
j 1
J
m
j 1 h 1
k = 1, …, K; m = 1, …, M
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(5)
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Model (5) is a constrained nonlinear problem. In general, a commercial software package, such as Lingo, can be used to solve it. Summing the CRs’ importance scores, the fuzzy customer satisfaction of QFD for NPP can be
expressed as:
Yˆ
L
I
k yˆ i 1
L
i
i
and
(6a)
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Yˆ
U
I
k yˆ i 1
U
i
(6b)
i
where ki is the importance score of CRi, i=1, …, I;
k
i
1 . The outcomes of Eqs. (6a) and (6b)
are set as the maximization objective for NPP. The FGP models used to achieve the optimal
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satisfaction of all goals in terms of the fulfillment levels of DRs are introduced in the next section.
3. Formulations
This section introduces two FGP models for NPP. The concept of -cuts and the extension principle are applied to transform the FGP models into conventional linear programming models to
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determine the fulfillment levels of DRs that achieve the maximum satisfaction level of all goals. Customer satisfaction complies with Herzberg’s two-factor theory [35], and thus a modified FGP model that takes motivation and hygiene factors into account is used to further enhance customer satisfaction.
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3.1 FGP models
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The FGP model considers three objectives, namely the maximum customer satisfaction, minimum incremental cost, and minimum technical difficulty of DRs to assist DMs in determining
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the fulfillment levels of DRs to obtain the maximum sum of the satisfaction of objectives. According to the CR satisfaction model, the objective functions for maximizing customer
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satisfaction can be defined as follows: (7)
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I J J J max ki ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i ) i 1 j 1 j 1 h 1
~ where x (ji ) represents the decision variable, x (ji ) [0, 1]. Let C j and T j be the fuzzy
incremental cost and technical difficulty of achieving the fulfillment level of DRj, respectively. The business competition threshold j and the maximum possible ability j of DRj are considered as constraints for NPP. Furthermore, since the maximum satisfaction degree of each CR is 100%,
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ACCEPTED MANUSCRIPT the FGP model should be subject to the constraint that the satisfaction level of each CR cannot be greater than 1. Therefore, the FGP model is formulated as follows: I J J J max ki ˆ0(i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh(i ) i 1 j 1 j 1 h 1 J
min C j x j j 1 J
min T j x j
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j 1
s. t. J
J
J
ˆ0( i ) ˆ (j i ) Rij x (ji ) ˆ (jhi ) rjh x (ji ) xh( i ) 1, i j 1
0
j
j 1 h 1
xj
j
1, j
(8)
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To deal with the above FGP model, this study adopts Tiwari et al.’s idea [30] of using the additive model to aggregate the fuzzy goals. First, the aspiration levels for each goal are determined by the DM. Then the membership functions of the satisfaction degree of max-goal and min-goal can
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be formulated as follows, respectively [42, 43]:
L 0, if Gmax ( x) Gmax , L ( x) Gmax G L U max (x) maxU , if Gmax Gmax ( x) Gmax , L Gmax Gmax U 1 , if Gmax ( x) Gmax .
PT
ED
(9)
U 0, if Gmin ( x) Gmin , U Gmin ( x) G L U min (x) minU , if Gmin Gmin ( x) Gmin , L G G min min L 1, if Gmin ( x) Gmin ,
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CE
(10)
where x denotes the variable vector. Gmax (x) and Gmin (x) represent the satisfaction levels of max-goal and min-goal at x, respectively. Corresponding to Eq. (8), Gmax (x) represents J J ˆ (i ) J ˆ (i ) (i ) k R x ˆ (jhi ) rjh x(ji ) xh(i ) , while Gmin (x) denotes both i 0 j ij j i 1 j 1 j 1 h 1 I
J
C j x j and j 1
J
T x j 1
j
j
.
In Eqs. (9) and (10), L and U respectively specify the lower and upper bounds of the aspiration of the goal. Moreover, the preemptive priority structure is considered in the model. If three fuzzy goals 11
ACCEPTED MANUSCRIPT are considered as that the customer satisfaction (C. S.) is more important m times than the incremental cost expenditure, the preemptive priority structure can be denoted as membership functions and represented as C. S. (x) m Cost (x) , m>1. Therefore, Eq. (8) can be expressed as an additive model with a preemptive priority structure of goals and relevant constraints. Let 1 (x) ,
2 (x) , and 3 (x) represent the membership functions of satisfaction degrees of the customer
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satisfaction, cost, and technical difficulty, respectively. In addition, let p (x) , p, be greater than a threshold to ensure that none of the goals is ignored during the calculation. Combining the approaches above with the -cuts and extension principle in fuzzy sets [36, 37], Eq. (8) can be
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transformed into conventional linear programming models to determine the fulfillment levels of DRs that achieve the maximum satisfaction level of all goals. The membership function of the objective value Z in the linear programming model can be defined as
3
Z ( Z ) sup min R ( Rij ), r (rjh ), C (C j ), T (T j ), i, j, h Z p ( x)
jh
ij
j
j
p 1
(11)
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R , r ,C ,T
where R, r, C, and T are the element values of the fuzzy coefficients defined in their corresponding
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linguistic scale. Applying Eq. (11), Z ( Z ) can be determined using the membership levels of all
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fuzzy coefficients. Therefore, Eq. (8) can be simplified to conventional linear programming models and separated into two crisp sub-additive models to find the lower and upper bounds of membership
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functions of the satisfaction degree, shown as Eqs. (12a) and (12b), respectively: 3
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ZL max p ( x) p 1
s. t.
1 ( x)
I
J
i 1
j 1
J
j 1 h 1
U L GC. S. GC. S. J
2 ( x)
J
ki ( ˆ0(i ) ˆ (j i ) ( Rij )L x (ji ) ˆ (jhi ) (rjh )L x (ji ) xh( i ) ) GC.L S. U GCost (C j )L x j j 1
U L GCost GCost
,
12
,
ACCEPTED MANUSCRIPT J
3 ( x )
U L GT. D. (T j ) x j j 1
,
L GT. D.
U T. D.
G
J
J
J
(i ) ˆ0( i ) ˆ (j i ) ( Rij )L x (ji ) ˆ jh ( rjh )L x (ji ) xh( i ) 1, i, j 1
j 1 h 1
t ( x) mv ( x), m 1, t , v {1, 2, 3},
(12a)
t v
0
j
xj
1, j ,
j
3
ZU max p ( x) p 1
s. t. J
i 1
j 1
J
U C. S.
G J
2 ( x)
U GCost (C j )U x j j 1 U Cost
,
L GCost
G
J
U U GT. D. (T j ) x j j 1
U T. D. J
G
J
,
L GT. D.
J
j 1 h 1 L C. S.
G
,
M
3 ( x )
J
ki ( ˆ0(i ) ˆ (j i ) ( Rij )U x (ji ) ˆ (jhi ) (rjh )U x(ji ) xh( i ) ) GC.L S.
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1 ( x)
I
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p ( x) 1, p.
ˆ0(i ) ˆ (j i ) ( Rij )U x (ji ) ˆ jh(i ) (rjh )U x (ji ) xh(i ) 1, i, j 1
j 1 h 1
t (x) mv (x), m 1, t , v {1, 2, 3}, 0 j x j j 1, j ,
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p (x) 1, p.
(12b)
ED
t v
levels.
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The membership function of Z can then be constructed using various [ ZL , ZU ] from different
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3.2 FGP models with Herzberg’s two-factor theory This study employs Herzberg’s two-factor theory to increase customer satisfaction.
Considering the fulfillment levels of DRs from Eqs. (12a) and (12b), DRs with the lower fulfillment levels comply with the motivation factor in Herzberg’s two-factor theory [35]. This means that when a DR’s fulfillment level is increased, customer satisfaction will increase, but decreasing a DR’s fulfillment level will not have a great impact on customer satisfaction. In contrast, DRs with
13
ACCEPTED MANUSCRIPT higher fulfillment level comply with the hygiene factor in the two-factor theory. Increasing a DR’s fulfillment level will not enhance customer satisfaction significantly, but decreasing it will impact customer satisfaction. Therefore, a modified FGP model is proposed based on Eqs. (12a) and (12b), and the concept of Herzberg’s two-factor theory to enhance customer satisfaction for NPP as follows: 3
p 1
s. t. I
1 ( x)
J
k ( ˆ
(i ) 0
i
i 1
J
J
L ˆ (j i ) ( Rij )L x qj ( i ) ˆ (jhi ) ( rjh )L x qj ( i ) xhq ( i ) ) GC. S. j 1
U C. S.
G
j 1 h 1 L C. S.
G
J
U GCost (C j )L x qj j 1 U Cost
,
L GCost
G
J
3 ( x )
U L q GT. D. (T j ) x j j 1
U T. D.
G
,
L GT. D.
J
J
,
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2 ( x)
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ZL max p ( x)
J
M
ˆ0( i ) ˆ (j i ) ( Rij )L x qj ( i ) ˆ (jhi ) ( rjh )L x qj ( i ) xhq ( i ) 1, i, j 1
j 1 h 1
t ( x) mv ( x), m 1, t , v {1, 2, 3}, t v j
x qj
j
1, j ,
(13a)
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p ( x) 1, p.
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0
3
ZU max p ( x) s. t.
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p 1
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1 ( x)
2 ( x)
I
J
i 1
j 1
J
J
ki ( ˆ0(i ) ˆ (j i ) ( Rij )U x qj ( i ) ˆ (jhi ) (rjh )U x qj ( i ) xhq ( i ) ) GC.U S. U C. S.
G
j 1 h 1 L C. S.
G
J
U GCost (C j )U x qj j 1 U Cost
G
,
L GCost
J
3 ( x )
U U q GT. D. (T j ) x j
J
j 1
U T. D.
G
,
L GT. D. J
J
ˆ0( i ) ˆ (j i ) ( Rij )U x qj ( i ) ˆ (jhi ) ( rjh )U x qj ( i ) xhq ( i ) 1, i, j 1
j 1 h 1
14
,
ACCEPTED MANUSCRIPT t ( x) mv ( x), m 1, t , v {1, 2, 3}, t v
0
j
x qj
j
1, j ,
(13b)
p (x) 1, p, where q =0.5 or 1 if the corresponding DR is viewed as a motivation or hygiene factor, respectively. Of note, in Eqs. (13a) and (13b), the fulfillment level of DRj, j=1, 2, …, J, that is a motivation factor will be increased, whereas that of DRj that is a hygiene factor will be unchanged to increase
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customer satisfaction in the model.
4. Illustration
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A semiconductor packing case is used to illustrate the feasibility and applicability of the proposed models. In this section, the proposed model is applied to a turbo thermal ball grid array (T2-BGA). T2-BGA is a member of the plastic ball grid array (PBGA) family. Since a copper heat slug is lodged in the molding compound of PBGA, T2-BGA has good heat dissipation and electrical
M
characteristics. However, T2-BGA still has some issues since its structure is more complex than that of PBGA [44]. The cross-sections of PBGA and T2-BGA are shown in Fig. 2.
ED
[Insert Figure 2 around here]
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According to the characteristics of T2-BGA, the QFD team collects five CRs, namely ―package profile‖ (CR1), ―thermal performance‖ (CR2), ―electrical performance‖ (CR3), ―reliability‖ (CR4),
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and ―co-planarity‖ (CR5). Considering these CRs, the QFD team develops the corresponding DRs using new product feasibility studies. Five DRs are determined, namely ―heat slug exposed area‖
AC
(DR1), ―heat slug attached material‖ (DR2), ―height of heat slug‖ (DR3), ―copper pattern‖ (DR4), and ―molding flow‖ (DR5). An HOQ for the CRs and DRs above is determined, as shown in Fig. 3. The fuzzy relationships and correlations are denoted as ―very weak (VW)‖, ―weak (W)‖, ―moderate (M)‖, ―high (H)‖, or ―very high (VH)‖, which are converted into triangular fuzzy numbers as the three-element sets (0, 0, .2), (0, 0.2, 0.4), (0.3, 0.5, 0.7), (0.6, 0.8, 1), and (0.8, 1, 1), respectively. The middle value in the three-element set represents the most likely one to have a membership
15
ACCEPTED MANUSCRIPT degree equivalent to 1, such as ( M ) (0.5) 1 . The membership functions are shown in Fig. 4. For instance, for Rij = ―M‖, the membership function is defined as: ( Rij 0.3) , 0.3 R1,i 0.5 (0.5 0.3) Rij ( Rij ) (0.7 Rij ) , 0.5 R 0.7 1,i (0.7 0.5)
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[Insert Figures 3 and 4 around here]
To obtain each CR satisfaction degree model using Eqs. (2a) and (2b), the lower and upper bounds of each -cut of Rij and rj should be determined beforehand. The lower and upper
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bounds of each -cut can be represented as functions of . For example, the lower and upper bounds of each -cut of R1,2 can be expressed as [( R1,2 )L ,( R1,2 )U ] = [0.3 0.2 , 0 .7 0.2 ] . Let
e denote the eth level, [0, 1], where e 1 , e e E , and e e1 1 E , e {0,1, , E} .
M
This means that the distance between two arbitrarily adjacent levels is equal in [0, 1]. 4.1 Fuzzy customer satisfaction model of CR
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To collect the input-output data set, the CR satisfaction degree, in the fuzzy sense, and the
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corresponding DRs’ fulfillment levels are defined as the response variable and explanatory variables, respectively. The fuzzy satisfaction degrees are denoted as ―not satisfied (NS)‖, ―slightly satisfied
CE
(SS)‖, ―medium satisfied (MS)‖, ―satisfied (S)‖ or ―very satisfied (VS)‖, which are converted to triangular fuzzy numbers as the 3-element sets (0, 0, 0.3), (0, 0.25, 0.5), (0.3, 0.5, 0.7), (0.5, 0.75,
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1.0), and (0.7, 1.0, 1.0), respectively. Their membership functions are shown in Fig. 5. According to the HOQ of T2-BGA in Fig. 3, the QFD team employs experimental design to construct the initial input-output data sets by considering various portfolios between each CR’s fuzzy satisfaction degree and the corresponding DRs’ fulfillment levels based on the QFD team’s experience and domain knowledge. The initial input-output data sets for the CRs are listed in Tables 1 to 5. Each sample in these tables can be considered a different confidence level (i.e. -level) of the fuzzy
16
ACCEPTED MANUSCRIPT satisfaction degree. In this study, three levels are adopted, {0.2, 0.8, 1}, for each sample such that five satisfaction degrees (output data) are collected from each sample in Tables 1 to 5. The data of DR fulfillment levels in Tables 1 to 5 have to be combined with their fuzzy relationships and correlations (shown in Fig. 3) with the same confidence levels to determine the corresponding input data for formulating each CR’s satisfaction model. Employing Eqs. (4) and (5), the vectors of the
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estimators of Eq. (3), ˆ ( i ) , i=1, …, 5, can be obtained. The predicted fuzzy satisfaction degree expression of each CR can be expressed as:
yˆ1 0.43R12 x2 0.381R13 x3 2.021R15 x5 0.118r23 x2 x3 0.542r25 x2 x5
(14-1)
yˆ 2 1.333R21 x1 2.611R24 x4
(14-2)
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yˆ3 1.25R31 x1 1.318R34 x4
(14-3) (14-4)
yˆ5 0.24 2.5R54 x4
(14-5)
M
yˆ4 0.037 0.599R42 x2 2.132R45 x5 0r25 x2 x5
The predicted fuzzy satisfaction degree expression for each CR (Eqs. (14-1) to (14-5)) can be
ED
expressed by the lower and upper bounds of the level, as done in Eqs. (3a) and (3b). Suppose that the importance score of each DR is determined by the QFD team as {0.14, 0.24, 0.17, 0.32, 0.13}.
CE
then be obtained.
PT
Using Eqs. (6a) and (6b), the lower and upper bounds of the fuzzy customer satisfaction model can
[Insert Figure 5 around here]
[Insert Tables 1 and 5 around here]
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4.2 FGP models
Referring to Eq. (8), the QFD team determines the fuzzy incremental cost and technical
difficulty for each DR, which are listed in Table 6. To transform the FGP model into a linear programming model, the lower and upper bounds of the aspiration levels of the three goals are set as shown in Fig. 6. Based on Table 6 and Fig. 6, the membership functions of satisfaction degree of customer satisfaction, incremental cost, and technical difficulty can be formulated using Eqs. (9)
17
ACCEPTED MANUSCRIPT and (10) as follows:
C. S. (x) [(0.043 0.1065 x1 0.1834 x2 0.0267 x3 0.3024 x4 0.193x5 0.0132 x2 x3 0.0152 x2 x5 ) 0.3] / 4
Cost (x) [1.8 (0.25 x1 0.75 x2 0.5 x3 0.5 x4 0.25 x5 )] / 0.8
(15) (16)
T. D. (x) [2.0 (0.75 x1 0.25 x2 0.5 x3 0.25 x4 0.75 x5 )] /1.2 (17)
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Considering the preemptive priority structure of goals, suppose that the goal of the customer satisfaction is 1.05 times more important than those of incremental cost and technical difficulty. The threshold is set to 0.05 for each goal. The business competition threshold j and the maximum possible ability j of DRj , j = 1,…,5, are set as 0.2 and 0.95, respectively. According to the
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information above, the various lower and upper membership levels of fuzzy goals and the fulfillment levels of DRs can be obtained using Eqs. (12a) and (12b). Table 7 lists the outcomes of Eqs. (12a) and (12b) for the present case.
M
[Insert Figure 6 around here] [Insert Table 6 around here] [Insert Table 7 around here] To enhance the customer satisfaction degree, the outcomes of the fulfillment levels of DRs in
ED
Table 7 are defuzzified and categorized into motivation and hygiene factors. A DR is categorized as a motivation factor if its defuzzified fulfillment level is less than 0.7; otherwise, it is classified as a
PT
hygiene factor. In this case, the fuzzy mean (FM) method [45] is adopted to perform the
CE
defuzzification of xi , i = 1,…, 5, since it is simple and efficient. F
f 1 F
(18)
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d
f xf f f 1
where f and x f denote the membership degree and the representative numerical value of the fth output, respectively. In this study, d is the defuzzification value of interval values [ x Lf , x Uf ], i.e., the fth -cut of the fuzzy number. Applying Mabuchi’s idea [46], the representative value of the fth
-cut is defined as the average of the lower and upper bounds of the interval: x f 1 2 ( x Lf x Uf ) . 18
ACCEPTED MANUSCRIPT Based on the FM method and Mabuchi’s idea, the fulfillment levels of DR1, …, DR5 can be defuzzified as {0.2, 0.557, 0.2, 0.95, 0.708} based on an level set with {1, 0.95, 0.9}. Therefore, DR1, DR2, and DR3 are categorized as the motivation factors, and DR4 and DR5 are classified as hygiene factors. According to this DR categorization, the acceptable confidence degree must be greater than or equal to 0.5. The various lower and upper membership levels of fuzzy goals are
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updated and the modified fulfillment levels of DRs can be obtained using Eqs. (13a) and (13b). Table 8 lists the outcomes of Eqs. (13a) and (13b) for T2-BGA. In Table 8, the acceptable confidence degree produces infeasible regions at 0.5 in Eq. (13b) of the QFD processes. [Insert Table 8 around here]
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4.3 Discussion
In the present case, 90 observed data were collected using experimental design and fuzzy set theory based on the QFD team’s knowledge and experience. This approach is reasonable for NPP in a practical (fuzzy) environment. The formulations of each CR’s satisfaction degree are listed in Eqs
M
(14-1) to (14-5). They are useful for determining the models of total customer satisfaction degree in
ED
QFD processes.
For comparison, the customer satisfaction degree of models (13a) and (13b), as well as (12a)
PT
and (12b), were determined using Eqs. (6a) and (6b) based on the outcomes of the fulfillment levels of DRs in Tables 7 and 8. Models (13a) and (13b), as well as (12a) and (12b), include 24 constraints
CE
and 8 decision variables.
AC
Referring to Tables 7 and 8, the outcome of C.S. from models (13a) and (13b) is better than that from models (12a) and (12b) at all levels; on the other hand, the outcomes of Cost and
T. D. from models (13a) and (13b) are less than those from models (12a) and (12b) at all levels under the same constraints. The outcomes of C.S. , Cost and T. D. in Table 8 denote the weight assignments of the three goals. Figure 7 shows the customer satisfaction degree based on models (13a) and (13b), as well as (12a) and (12b), of the QFD processes for levels greater than or equal 19
ACCEPTED MANUSCRIPT to 0.9. The customer satisfaction degrees for models (13a) and (13b) (i.e., with two-factor theory idea, YTFT) are better than those for models (12a) and (12b) for NPP of T2-BGA. In this case, at = 1, the customer satisfaction degree for models (13a) or (13b) is enhanced by 14.1% compared to that for models (12a) or (12b) based on the outcomes of DR fulfillment levels in Tables 7 and 8. In Table 7 or 8, it is noted that maximizing the model may cause some decision variables at the
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lower bound to be greater than those at the upper bound for their corresponding objective functions. For example, in Tables 7, x5(L) is greater than x5(U) , at = 0.95 and x2(L) is greater than x2(U) at
= 0.9. However, the maximum membership degrees of all goals at the upper bound are greater than those at the lower bound for both proposed models. To avoid confusion, the optimal fulfillment
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levels of DRs determined for the lower and upper bounds of the membership degree of satisfaction degree are respectively denoted as (L) and (U) in the tables. Considering the fuzziness of NPP, the lower acceptable confidence levels may produce infeasible regions when the proposed models are
M
solved. Once the infeasible regions are found, the DM has to determine a higher acceptable confidence degree to reduce uncertainty for obtaining adequate decision-making information for
ED
NPP. Models (5), (12a) and (12b), (13a) and (13b) were solved using Lingo 10 software in a Microsoft Windows 7 environment running on a laptop with a 2.1-GHz processor and 2 GB random
CE
calculation.
PT
access memory. The efficiency was satisfactory, with an almost negligible runtime for each
Although QFD was originally developed for product improvements or NPP, it has been
AC
successfully applied in various fields and industries [1-5, 47-50], as QFD frameworks are an effective approach for describing the relationships between ―what is needed‖ and ―how to do‖ in problems. As such, even though the proposed approaches are developed for NPP in this study, they can be easily applied to other fields or problems. The input-output observation data set was collected by carrying out an experiment design. To obtain legitimate observations, QFD team members should be capable of providing the corresponding fuzzy responses based on the
20
ACCEPTED MANUSCRIPT combinations of fulfillment levels of corresponding DRs. Inappropriate fuzzy responses will result in incorrect model-fittings for CRs and the subsequent decision for customer satisfaction.
5. Conclusion Determining the fulfillment levels of DRs to meet CRs to achieve optimal customer
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satisfaction is a major task in NPP. QFD is a systematic tool for NPP. This study considered three objectives, namely the maximum customer satisfaction, minimum incremental cost, and minimum technical difficulty, of DRs for NPP in a fuzzy environment. Applying the FGP approach and considering the preemptive priority of the various goals, each CR’s optimal satisfaction can be
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attained based on the overall goals in terms of the fulfillment levels of DRs. Each CR’s satisfaction level and the fulfillment levels of the corresponding DRs are defined as the response variable and explanatory variables, respectively. Each CR’s satisfaction expression is determined using the mathematical programming method. The observed data of the response and explanatory variables
M
are collected from the alternatives in the competition analysis. Experimental design and fuzzy sets
ED
are employed to collect the input-output data set based on the evaluated relationships between CRs and DRs and the correlations among DRs in QFD processes using the QFD team’s knowledge and
PT
experience. In addition, Herzberg’s two-factor theory is incorporated into the models of customer satisfaction. Updated models were proposed to enhance customer satisfaction for NPP.
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The proposed models were illustrated with a numerical example to demonstrate their
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applicability in practice. The resulting ranges of satisfaction degrees and the possible ranges of fulfillment levels of DRs in both models can provide the QFD team with useful information for new product design. The model that considers two-factor theory produced better outcomes of customer satisfaction of all goals. In future research, multiple segmentations of the market and the integration of the more ―what‖-―how‖ information of QFD activities can be considered for developing more sophisticated models.
21
ACCEPTED MANUSCRIPT Acknowledgement This research was funded by Contract NSC 98-2410-H-006-042-MY3 and in part by NSC 101-2410-H-168-003 from the Ministry of Science and Technology, Republic of China.
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Captions for figures and tables Figure 1. House of quality. Figure 2. Cross-section diagrams of PBGA and T2-BGA. Figure 3. HOQ for T2BGA. Figure 4. Membership functions of linguistic terms for relationships and correlations in HOQ. Figure 5. Membership functions of linguistic terms for ranking of CR satisfaction levels. 26
ACCEPTED MANUSCRIPT Figure 6. Lower and upper bounds of aspiration levels of three goals. Figure 7. Membership functions of customer satisfaction degree based on models (12) and (13). Table 1. Initial input-output data sets of CR1 and DRs. Table 2. Initial input-output data sets of CR2 and DRs. Table 3. Initial input-output data sets of CR3 and DRs.
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Table 4 Initial input-output data sets of CR4 and DRs. Table 5 Initial input-output data sets of CR5 and DR4.
Table 6. Fuzzy incremental cost and technical difficulty for each DR
Table 7 Outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various levels.
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Table 8 Updated outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various levels.
rjJ
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correlation among DRs
ED
CR1 k1 . . . . . .
Importance scores of CRs
DR1 . . . DRj . . . DRJ
ki . . .
CRI
kI
AC
CE
PT
CRi . . .
Relationship between CRi and DRj Rij
Importance ratings of DRs W1 . . . Wj . . . WJ
Figure 1. House of quality.
27
ACCEPTED MANUSCRIPT Chip
Gold wire
Heat slug
Solder ball
Molding compound
T2 - BGA
PBGA
Substrate
W H DR1 DR2 DR3 DR4 DR5 CR1 CR2
W
CR3
W
W
M
AN US
M
W M
H
CR4
CR IP T
Figure 2.Cross-section diagrams of PBGA and T2-BGA.
W
W
M
CR5
1 VW W
M
AC
CE
PT
Membership degree
ED
Figure 3. HOQ for T2BGA.
0
H
.2 .3 .4 .5 .6 .7 .8
VH
1
Linguistic terms
Figure 4. Membership functions of linguistic terms for relationships and correlations in HOQ.
28
ACCEPTED MANUSCRIPT NS
MS
SS
S
VS
Membership degree
1
0 1
2
3
4
5
6
7
8
9
10
CR IP T
0
0
Membership degree
0
0.7 GC .S . (x)
1
1.8 GCost (x)
1
0
M
0.3
1
AN US
1
Membership degree
Membership degree
Figure 5. Membership functions of linguistic terms for ranking of CR satisfaction levels.
0.8
2.0 GT . D. (x)
ED
Figure 6. Lower and upper bounds of aspiration levels of three goals.
AC
CE
Membership degree
PT
1
Yˆ
Yˆ TFT
0.99
0.98 0.58
0.6 0.62 0.64 0.66 0.68 0.7 Customer satisfaction degree
Figure 7. Membership functions of customer satisfaction degree based on models (12) and (13).
29
ACCEPTED MANUSCRIPT Table 1. Initial input-output data sets of CR1 and DRs. Satisfaction level of CR1
Fulfillment levels of DR2, DR3, and DR5 Sample x3
x5
x2 x 3
x2 x 5
y1
1 2 3 4 5 6 7
0.00 0.00 0.00 0.50 0.50 0.50 1.00
0.00 0.50 1.00 0.00 0.50 1.00 0.00
0.00 0.50 1.00 0.00 0.50 1.00 0.50
0.00 0.00 0.00 0.00 0.25 0.50 0.00
0.00 0.00 0.00 0.00 0.25 0.50 0.50
NS MS S NS MS VS MS
8 9 10 11 12 13 14 15 16 17
1.00 1.00 0.00 0.00 0.00 0.50 0.50 0.50 1.00 1.00
0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50
1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00
0.50 1.00 0.00 0.00 0.00 0.00 0.25 0.50 0.00 0.50
1.00 0.00 0.00 0.00 0.00 0.25 0.50 0.00 0.50 1.00
18
1.00
1.00
0.00
1.00
0.00
CR IP T
x2
MS
ED
M
AN US
VS MS NS MS NS SS S SS MS VS
PT
Table 2. Initial input-output data sets of CR2 and DRs. Fulfillment levels of DR1 and DR4
Satisfaction level of CR2
Sample
x4
y2
1 2
0 0
0 0.5
NS SS
3 4 5 6 7 8 9 10 11
0 0.5 0.5 0.5 1 1 1 0 0
1 0 0.5 1 0 0.5 1 0 0.5
MS NS MS S SS S VS NS SS
AC
CE
x1
30
ACCEPTED MANUSCRIPT 12 13
0 0.5
1 0
MS NS
14 15 16 17 18
0.5 0.5 1 1 1
0.5 1 0 0.5 1
MS S SS S VS
Fulfillment levels of DR1 and DR4 Sample x4
1 2
0 0
0 0.5
3 4 5 6 7 8
0 0.5 0.5 0.5 1 1
9 10 11 12 13 14 15 16 17 18
1 0 0 0 0.5 0.5 0.5 1 1 1
y3
NS SS
1 0 0.5 1 0 0.5
MS NS MS VS SS S
1 0 0.5 1 0 0.5 1 0 0.5 1
VS NS SS MS NS MS VS SS S VS
M
ED PT
CE AC
Satisfaction level of CR3
AN US
x1
CR IP T
Table 3. Initial input-output data sets of CR3 and DRs.
Table 4. Initial input-output data sets of CR4 and DRs. Fulfillment levels of DR2 and DR5
Satisfaction level of CR4
x2
x5
x2 x 5
y4
0.00 0.50
0.00 0.50
0.00 0.25
NS MS
Sample 1 2
31
ACCEPTED MANUSCRIPT 1.00 0.00
1.00 0.00
1.00 0.00
VS NS
5 6 7 8 9 10 11 12 13
0.50 1.00 0.00 0.50 1.00 0.00 0.50 1.00 0.00
0.50 1.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50
0.25 1.00 0.00 0.50 0.00 0.00 0.50 0.00 0.00
MS VS SS S MS SS S MS SS
14 15 16 17 18
0.50 1.00 0.00 0.50 1.00
1.00 0.00 0.50 1.00 0.00
0.50 0.00 0.00 0.50 0.00
AN US
CR IP T
3 4
S MS SS S MS
Table 5. Initial input-output data sets of CR5 and DR4. Fulfillment levels of DR4
AC
CE 10 11 12 13 14 15 16 17 18
Satisfaction level of CR4
x4
y5
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
NS MS VS NS MS S NS MS S
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
NS MS S NS MS VS NS MS S
ED
PT
1 2 3 4 5 6 7 8 9
M
Sample
32
ACCEPTED MANUSCRIPT Table 6. Fuzzy incremental cost and technical difficulty for each DR DRj
Cj
-cut of C j
DR1
(0, 0.25, 0.5)
[0.25, 0.5-0.25]
-cut of T j
Tj
(0.5, 0.75, 1) [0.5+0.25, 1-0.25]
DR2 (0.5, 0.75, 1.) [0.5+0.25, 1-0.25] (0, 0.25, 0.5)
[0.25, 0.5-0.25]
DR3 (0.3, 0.5, 0.7) [0.3+0.2, 0.7-0.3] (0.3, 0.5, 0.7) [0.3+0.1, 0.7-0.3] DR4 (0.3, 0.5, 0.7) [0.3+0.2, 0.7-0.3] (0, 0.25, 0.5)
[0.25, 0.5-0.25]
[0.25, 0.5-0.25]
(0.5, 0.75, 1) [0.5+0.25, 1-0.25]
CR IP T
DR5
(0, 0.25, 0.5)
Table 7 Outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various levels.
L U L U L U C.S. C.S. Cost Cost T.D. T.D. x1(L) x1(U) x2(L)
0.95 0.709 0.794 0.676 0.756 0.676 0.756 0.2 0.9 0.666 0.889 0.635 0.847 0.635 0.692 0.2
x3(L)
0.2 0.576 0.576 0.2
x3(U)
x4(L)
x4(U)
x5(L)
x5(U)
0.2 0.95 0.95 0.678 0.678
AN US
1 0.752 0.752 0.717 0.717 0.717 0.717 0.2
x2(U)
0.2 0.572 0.58 0.2
0.2 0.95 0.95 0.701 0.655
0.2 0.568 0.464 0.2
0.2 0.95 0.95 0.725 0.824
M
Table 8 Updated outcomes of goals’ satisfaction degrees and DRs’ fulfillment levels at various
1
L U L U L U C.S. C.S. Cost Cost T.D. T.D.
x1L
x1U
x2L
x2U
x3L
x3U
x4L
x4U
x5L
x5U
0.964 0.964 0.19 0.19 0.209 0.209 0.447 0.447 0.447 0.447 0.447 0.447 0.95 0.95 0.678 0.678
PT
ED
levels.
- 0.145 - 0.186
-
0.447
-
AC
0.95 0.893
CE
0.98 0.936 0.9930.1720.208 0.2 0.218 0.447 0.447 0.447 0.447 0.447 0.447 0.95 0.95 0.669 0.687
33
0.447
-
0.447
-
0.95
-
0.655
-